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For a general ${\cal N}=1$ supersymmetric gauge theory regularized by highercovariant derivatives we prove in all orders that the $\beta$-function definedin terms of the bare couplings is given by integrals of double totalderivatives with respect to loop momenta. With the help of the technique usedfor this proof it is possible to construct a method for obtaining these loopintegrals, which essentially simplifies the calculations. As an illustration ofthis method, we find the expression for the three-loop contribution to the$\beta$-function containing the Yukawa couplings and compare it with the resultof the standard calculations made earlier. Also we briefly discuss, how thestructure of the loop integrals for the $\beta$-function considered in thispaper can be used for the all-loop perturbative derivation of the NSVZ relationin the non-Abelian case.

the journal of high energy physics/the journal of high energy physics

The $\beta$-function of ${\cal N}=1$ supersymmetric gauge theories  regularized by higher covariant derivatives as an integral of double total  derivatives